Finite Math Examples

$\frac{x^{2}}{48} + \frac{y^{2}}{64} = 1$

Step 1

Subtract $\frac{x^{2}}{48}$ from both sides of the equation.

$\frac{y^{2}}{64} = 1 - \frac{x^{2}}{48}$

Step 2

Multiply both sides of the equation by $64$ .

$64 \frac{y^{2}}{64} = 64 (1 - \frac{x^{2}}{48})$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $64$ .

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Step 3.1.1.1

Cancel the common factor.

$64 \frac{y^{2}}{64} = 64 (1 - \frac{x^{2}}{48})$

Step 3.1.1.2

Rewrite the expression.

$y^{2} = 64 (1 - \frac{x^{2}}{48})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $64 (1 - \frac{x^{2}}{48})$ .

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Step 3.2.1.1

Apply the distributive property.

$y^{2} = 64 \cdot 1 + 64 (- \frac{x^{2}}{48})$

Step 3.2.1.2

Multiply $64$ by $1$ .

$y^{2} = 64 + 64 (- \frac{x^{2}}{48})$

Step 3.2.1.3

Cancel the common factor of $16$ .

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Step 3.2.1.3.1

Move the leading negative in $- \frac{x^{2}}{48}$ into the numerator.

$y^{2} = 64 + 64 \frac{- x^{2}}{48}$

Step 3.2.1.3.2

Factor $16$ out of $64$ .

$y^{2} = 64 + 16 (4) \frac{- x^{2}}{48}$

Step 3.2.1.3.3

Factor $16$ out of $48$ .

$y^{2} = 64 + 16 \cdot 4 \frac{- x^{2}}{16 \cdot 3}$

Step 3.2.1.3.4

Cancel the common factor.

$y^{2} = 64 + 16 \cdot 4 \frac{- x^{2}}{16 \cdot 3}$

Step 3.2.1.3.5

Rewrite the expression.

$y^{2} = 64 + 4 \frac{- x^{2}}{3}$

Step 3.2.1.4

Combine $4$ and $\frac{- x^{2}}{3}$ .

$y^{2} = 64 + \frac{4 (- x^{2})}{3}$

Step 3.2.1.5

Simplify the expression.

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Step 3.2.1.5.1

Multiply $- 1$ by $4$ .

$y^{2} = 64 + \frac{- 4 x^{2}}{3}$

Step 3.2.1.5.2

Move the negative in front of the fraction.

$y^{2} = 64 - \frac{4 x^{2}}{3}$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{64 - \frac{4 x^{2}}{3}}$

Step 5

Simplify $\pm \sqrt{64 - \frac{4 x^{2}}{3}}$ .

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Step 5.1

To write $64$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{64 \cdot \frac{3}{3} - \frac{4 x^{2}}{3}}$

Step 5.2

Simplify terms.

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Step 5.2.1

Combine $64$ and $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{64 \cdot 3}{3} - \frac{4 x^{2}}{3}}$

Step 5.2.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{64 \cdot 3 - 4 x^{2}}{3}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Factor $4$ out of $64 \cdot 3 - 4 x^{2}$ .

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Step 5.3.1.1

Factor $4$ out of $64 \cdot 3$ .

$y = \pm \sqrt{\frac{4 (16 \cdot 3) - 4 x^{2}}{3}}$

Step 5.3.1.2

Factor $4$ out of $- 4 x^{2}$ .

$y = \pm \sqrt{\frac{4 (16 \cdot 3) + 4 (- x^{2})}{3}}$

Step 5.3.1.3

Factor $4$ out of $4 (16 \cdot 3) + 4 (- x^{2})$ .

$y = \pm \sqrt{\frac{4 (16 \cdot 3 - x^{2})}{3}}$

Step 5.3.2

Multiply $16$ by $3$ .

$y = \pm \sqrt{\frac{4 (48 - x^{2})}{3}}$

Step 5.4

Rewrite $\sqrt{\frac{4 (48 - x^{2})}{3}}$ as $\frac{\sqrt{4 (48 - x^{2})}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{4 (48 - x^{2})}}{\sqrt{3}}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Rewrite $4$ as $2^{2}$ .

$y = \pm \frac{\sqrt{2^{2} (48 - x^{2})}}{\sqrt{3}}$

Step 5.5.2

Pull terms out from under the radical.

$y = \pm \frac{2 \sqrt{48 - x^{2}}}{\sqrt{3}}$

Step 5.6

Multiply $\frac{2 \sqrt{48 - x^{2}}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{2 \sqrt{48 - x^{2}}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.7

Combine and simplify the denominator.

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Step 5.7.1

Multiply $\frac{2 \sqrt{48 - x^{2}}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 5.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 5.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.7.5

Add $1$ and $1$ .

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 5.7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.7.6.4

Cancel the common factor of $2$ .

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Step 5.7.6.4.1

Cancel the common factor.

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.7.6.4.2

Rewrite the expression.

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{3^{1}}$

Step 5.7.6.5

Evaluate the exponent.

$y = \pm \frac{2 \sqrt{48 - x^{2}} \sqrt{3}}{3}$

Step 5.8

Combine using the product rule for radicals.

$y = \pm \frac{2 \sqrt{3 (48 - x^{2})}}{3}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{2 \sqrt{3 (48 - x^{2})}}{3}$

Step 6.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{2 \sqrt{3 (48 - x^{2})}}{3}$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{2 \sqrt{3 (48 - x^{2})}}{3}$

$y = - \frac{2 \sqrt{3 (48 - x^{2})}}{3}$

$y = \frac{2 \sqrt{3 (48 - x^{2})}}{3}$

$y = - \frac{2 \sqrt{3 (48 - x^{2})}}{3}$

Step 7

Set the radicand in $\sqrt{3 (48 - x^{2})}$ greater than or equal to $0$ to find where the expression is defined.

$3 (48 - x^{2}) \geq 0$

Step 8

Solve for $x$ .

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Step 8.1

Divide each term in $3 (48 - x^{2}) \geq 0$ by $3$ and simplify.

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Step 8.1.1

Divide each term in $3 (48 - x^{2}) \geq 0$ by $3$ .

$\frac{3 (48 - x^{2})}{3} \geq \frac{0}{3}$

Step 8.1.2

Simplify the left side.

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Step 8.1.2.1

Cancel the common factor of $3$ .

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Step 8.1.2.1.1

Cancel the common factor.

$\frac{3 (48 - x^{2})}{3} \geq \frac{0}{3}$

Step 8.1.2.1.2

Divide $48 - x^{2}$ by $1$ .

$48 - x^{2} \geq \frac{0}{3}$

Step 8.1.3

Simplify the right side.

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Step 8.1.3.1

Divide $0$ by $3$ .

$48 - x^{2} \geq 0$

Step 8.2

Subtract $48$ from both sides of the inequality.

$- x^{2} \geq - 48$

Step 8.3

Divide each term in $- x^{2} \geq - 48$ by $- 1$ and simplify.

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Step 8.3.1

Divide each term in $- x^{2} \geq - 48$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 48}{- 1}$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 48}{- 1}$

Step 8.3.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 48}{- 1}$

Step 8.3.3

Simplify the right side.

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Step 8.3.3.1

Divide $- 48$ by $- 1$ .

$x^{2} \leq 48$

Step 8.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{48}$

Step 8.5

Simplify the equation.

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Step 8.5.1

Simplify the left side.

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Step 8.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{48}$

Step 8.5.2

Simplify the right side.

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Step 8.5.2.1

Simplify $\sqrt{48}$ .

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Step 8.5.2.1.1

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 8.5.2.1.1.1

Factor $16$ out of $48$ .

$| x | \leq \sqrt{16 (3)}$

Step 8.5.2.1.1.2

Rewrite $16$ as $4^{2}$ .

$| x | \leq \sqrt{4^{2} \cdot 3}$

Step 8.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 4 | \sqrt{3}$

Step 8.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$| x | \leq 4 \sqrt{3}$

Step 8.6

Write $| x | \leq 4 \sqrt{3}$ as a piecewise.

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Step 8.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 8.6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 4 \sqrt{3}$

Step 8.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 8.6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 4 \sqrt{3}$

Step 8.6.5

Write as a piecewise.

${\begin{cases} x \leq 4 \sqrt{3} & x \geq 0 \\ - x \leq 4 \sqrt{3} & x < 0 \end{cases}$

Step 8.7

Find the intersection of $x \leq 4 \sqrt{3}$ and $x \geq 0$ .

$0 \leq x \leq 4 \sqrt{3}$

Step 8.8

Solve $- x \leq 4 \sqrt{3}$ when $x < 0$ .

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Step 8.8.1

Divide each term in $- x \leq 4 \sqrt{3}$ by $- 1$ and simplify.

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Step 8.8.1.1

Divide each term in $- x \leq 4 \sqrt{3}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{4 \sqrt{3}}{- 1}$

Step 8.8.1.2

Simplify the left side.

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Step 8.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{4 \sqrt{3}}{- 1}$

Step 8.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{4 \sqrt{3}}{- 1}$

Step 8.8.1.3

Simplify the right side.

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Step 8.8.1.3.1

Move the negative one from the denominator of $\frac{4 \sqrt{3}}{- 1}$ .

$x \geq - 1 \cdot (4 \sqrt{3})$

Step 8.8.1.3.2

Rewrite $- 1 \cdot (4 \sqrt{3})$ as $- (4 \sqrt{3})$ .

$x \geq - (4 \sqrt{3})$

Step 8.8.1.3.3

Multiply $4$ by $- 1$ .

$x \geq - 4 \sqrt{3}$

Step 8.8.2

Find the intersection of $x \geq - 4 \sqrt{3}$ and $x < 0$ .

$- 4 \sqrt{3} \leq x < 0$

Step 8.9

Find the union of the solutions.

$- 4 \sqrt{3} \leq x \leq 4 \sqrt{3}$

Step 9

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 4 \sqrt{3}, 4 \sqrt{3}]$

Set-Builder Notation:

${x | - 4 \sqrt{3} \leq x \leq 4 \sqrt{3}}$

Step 10

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- 8, 8]$

Set-Builder Notation:

${y | - 8 \leq y \leq 8}$

Step 11

Determine the domain and range.

Domain: $[- 4 \sqrt{3}, 4 \sqrt{3}], {x | - 4 \sqrt{3} \leq x \leq 4 \sqrt{3}}$

Range: $[- 8, 8], {y | - 8 \leq y \leq 8}$

Step 12